If $P = \begin{bmatrix} 1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4 \end{bmatrix}$ is the adjoint of the $3 \times 3$ matrix $A$ and $\det(A) = 4$, then $\alpha$ is equal to

If matrix $A = \begin{bmatrix} 3 & 2 & 4 \\ 1 & 2 & -1 \\ 0 & 1 & 1 \end{bmatrix}$ and $A^{-1} = \frac{1}{K} \text{adj}(A)$,then $K$ is:

If $A$ and $B$ are two matrices given by $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 5 & 6 & 8 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 2 & 5 \\ 2 & 3 & 8 \\ 7 & 2 & 9 \end{bmatrix}$,then the value of $|\operatorname{Adj}(AB)|$ is

Let $A=\left[\begin{array}{ll}x & 1 \\ 1 & 0\end{array}\right]$,$x \in R^{+}$ and $A^4=\left[a_{ij}\right]_2$. If $a_{11}=109$,then $\left(A^4\right)^{-1}=$

If $A$ is a $3 \times 3$ matrix such that $|A|=27$ and $\operatorname{Adj}(A)=k A^T$, then find the value of $k^2-3 k+5$.

If $A=\left[\begin{array}{ccc}1 & -2 & 2 \\ 0 & 2 & -3 \\ 3 & -2 & 4\end{array}\right]$,then $A \cdot \operatorname{adj}(A)$ is equal to